1. Non-dimensional parameters that determine flapping wing aerodynamics.
Non-dimensional parameters that determine flapping wing aerodynamics. Three non-dimensional numbers that relate air, kinematic and morphology parameters to key flow phenomena in flapping animal flight – the Reynolds number (Re), advance ratio (J) and Rossby number (Ro) – can be interpreted visually as length scale ratios (Lentink and Dickinson, 2009). (A) Transition of the wing boundary layer from laminar to turbulent flow depends on Re, which is proportional to the square of the chord length, c, measured in boundary layer thickness, δ: Re∝(c/δ)2. At the high Re of vertebrates, the chord length is much larger than the boundary layer thickness – compare this with insects, which have relatively thick viscous boundary layers. Uf, forward velocity. Figure adapted from Shyy et al. (2013), with permission. (B) The stability of the leading edge vortex on a wing swinging around a ‘shoulder’ joint depends on Ro, which is equal to the wing radius, R, divided by chord length, c, during hovering flight when J=0 (the boundary layer on the lower side corresponds to A). Figure adapted from Sane (2003), with permission. (C) The interactions of a vortex wake with the flapping wing depend on J, which is equal to the forward distance traveled by the wing base, s, divided by the total distance traveled by the wingtip in the stroke plane, 2A, during one wingbeat. The wing is shown during mid-stroke, when a stable LEV is present (the cross-section corresponds to B). The LEV connects to the tip vortex and is shed after each half-stroke. J governs the resulting wake dynamics and the interactions of the wake with the wing. Uw, wingtip velocity in the stroke plane; ν, kinematic viscosity of the fluid. During forward flight, Ro increases with J, which explains why less-prominent LEVs are formed at high J in animal flight. Figure adapted from Lentink and Dickinson (2009), with permission.
Diana D. Chin, and David Lentink J Exp Biol 2016;219:
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